Obi can be used to add cloth simulation to skinned meshes too, such as -but not limited to- characters. Add an Obi Cloth component any object containing a Skinned Mesh Renderer, and set it up as usual (create an Obi Mesh Topology for the skinned mesh, and add the cloth actor to a solver). Initially all particles in a skinned Obi Cloth are fixed and will follow the mesh animation. You'll have to unfix those particles which you want to be simulated.
When simulating a skinned cloth, all skinned vertex positions are retrieved each frame. In order to get up to date vertex positions, the animation must be updated before Obi starts the simulation for that the frame, or Obi will work with values from the previous frame. In Unity, the Animator component by default updates the animation before LateUpdate, which is (no pun intended) too late. You should set the Update Mode property of your Animator to "Animate physics". If you don´t, the simulation will be delayed 1 frame respect to the animation.
Fixed particles are then directly moved to the position of their skinned vertices, and free particles are simulated as usual. Skin Constraints try to make sure free particles do not end up too far away from the position they'd be if they were completely skinned (fixed). This algorithm allows Obi to efficiently simulate character clothing while avoiding clipping, tunneling and similar artifacts.
You can use the particle editor to set per-particle values for the Skin Constraints, as explained here.
Tethers, bending, aerodynamics, collisions, etc. continue to work in the same way as with regular cloth. So you can, for instance, make your character's cape wave in the wind or collide with the environment.